The determinant `|(y^(2),-xy,x^(2)),(a,b,c),(a',b',c')|` is equal to

Options

A. `|{:(bx+ay,,cx+by),(b'x+a'y,,c'x+b'y):}|`
B. `|{:(ax+by,,bx+cy),(a'x+b'y,,b'x+c'y):}|`
C. `|{:(bx+cy,,ax+by),(b'x+c'y,,a'x+b'y):}|`
D. `|{:(ax+by,,bc+cy),(a'x+b'y,,b'x+c'y):}|`

Answer

D

Explanation

Let `Delta =|{:(y^(2),,-xy,,x^(2)),(a,,b,,c),(a',,b',,c'):}|`
Then
`Delta =(1)/(xy) |{:(xy^(2),,-xy,,x^(2)y),(ax,,b,,cy),(a'x,,b',,c'y):}| " " "[Applying "C_(1) to xC_(1),C_(3) to yC_(3)]`
`=(1)/(xy) |{:(0,,-xy,,0),(ax+by,,b,,bx+cy),(a'x+b'y,,b',,b'x+c'y):}|`
`"[Applying " C_(1) to C_(1) +yC_(2),C_(3) to C_(3) +xC_(2)]`
`=(1)/(xy) xy |{:(ax+by,,bx+cy),(a'x+b'y,,b'x+c'y):}|" ""[Expanding along " R_(1) "]"`
`=|{:(ax+by,,bx+cy),(a'x+b'y,,b'x+c'y):}|`